The microscopic world is deeply unintuitive, in large part due to the incessant randomness of Brownian motion. I was reminded of this when a colleague (Ben McMorran) pointed me to the timely topic of face masks and the physics that governs how they work. (Ben is collecting readings on this subject to discuss with undergraduates.) **How does the capture efficiency of a facemask depend on the size of the particles moving through it?** The answer is not at all obvious, though it’s simple in retrospect, and the question led to a nice discussion plus a homework problem I wrote for my biophysics class. I thought it worth writing up, in case it’s of use or of interest to others.

## The question

A cloth face mask is a fabric of fibers that passing particles might adhere to. Or, the particles might pass through. How does the likelihood of capture depend on a particle’s size?

First, as I asked my students to do, make a guess. Sketch a graph of what you imagine the capture efficiency vs. particle radius looks like. (*Do this!*)

Here are some shapes people sketched:

Before thinking about this, I probably would have guessed “*A*,” a monotonic function of size with larger objects being easier to capture than smaller ones.

In fact, the true curve is like “*B*,” with a minimum in the middle. I’ll show actual data below, but here’s a schematic diagram from a document by 3M:

Remarkably, the capture efficiency is non-monotonic. As particles get large, they’re easier to capture, but as they get very small, they’re also easier to capture.

The reason is **Brownian motion:** the unavoidable random meandering of particles is greater the smaller they are, enhancing their likelihood of hitting a fiber. Particles can also hit via simple impact, and this is more likely for larger particles. Therefore very small particles are well-captured, very large particles are well-captured, and there’s a regime in between at which we have a minimum of our capture efficiency. There’s a lot of physics in the details of the actual capture, but that doesn’t matter for the basic picture sketched here. These mechanisms, plus some neat long-range electrostatics, are qualitatively described in this excellent “Minute Physics” video.

## Your assignment, version 1

The great place to pause is here, and given the statement above, try **constructing a model yourself** that gives an equation for capture efficiency vs. particle radius, as a function of whatever parameters of masks, flows, and properties of nature you think are relevant. Then plot your equation, calculate where any minima are located, and plug in numbers to estimate the particle size that’s hardest to catch. (I did this; it was fun.) Assume that fibers are perfectly sticky; anything that hits them irrevocably adheres.

To encourage you to **pause and do this, **before sketching a model myself and unduly influencing your thoughts, I’ll break up the page with (*1*) a pointer to my book announcement page [link updated July 2021], and (*2*) this wonderful video of non-random motion, from Yoann Bourgeois & CCN2:

## Your assignment, version 2

This exercise spurred me to write a homework problem that provides a bit more structure. It goes like this:

Consider the diagram shown below, a schematic cross section of a mask with a particle approaching it from the left. The large circles are the fibers of the mask; suppose their diameter and edge-to-edge separation are both *L*. The small black circle is a spherical particle of radius *a*, moving to the right with velocity *v*. The mask extends to the right, to a total thickness *NL*. The dashes indicate streamlines (i.e. indicating how a massless point would flow).

Let’s *very roughly* estimate the capture efficiency of the mask — i.e. the probability that the particle will stick to the mask, rather than passing through. Suppose that any contact leads to irreversible adhesion [*]. There are two ways the particle can hit a fiber:

(1) **Impact: **leaving a streamline, since real particles have mass, and running into a fiber. Roughly, this probability p_{1} is proportional to the ratio of the cross-sectional area of the particle to the cross-sectional area of the fiber. (Ask yourself why this is reasonable.)

(2) **Diffusion:** meandering to a fiber via Brownian motion. Imagine a particle as illustrated; t_{a} is the time needed to travel “forward” by one fiber spacing; t_{D} is the typical time needed to diffuse “sideways” by one fiber spacing, thereby hitting a fiber. The probability for being captured is roughly p_{2} = t_{a} / t_{D} — i.e. if this ratio is small, we’re likely to pass through before wandering laterally and hitting a fiber.

**(a)** Does *N* matter? Why or why not? Suggestion: think separately whether the particle size dependence should involve *N*, and about whether the overall probability of capture from the whole fiber should involve *N*.

**(b)** Roughly, the total capture probability p = p_{1} + p_{2}. (Obviously, this is a decent approximation only for small p, and it has to “max out” at p=1; you can impose this ceiling by hand.) Write an expression for p as a function of *a, L*, the viscosity of air, temperature, and flow velocity. Keep in mind that this is a rough, order-of-magnitude estimate. Assess whether p has a maximum or minimum as a function of particle radius, and write an expression for this a value.

**(c) **Using 2 x 10^{-5} kg m^{-1} s^{-1} for the viscosity of air, *v* = 0.1 m/s and as a typical flow velocity of one’s breath, and *L* = 10 microns for the fiber diameter, plot p_{1}, p_{2}, and the total p vs. *a,* and calculate the numerical value of the *a* of any extremal point. (I suggest a log-log plot.)

**(d) **Compare your answer with the Figures 2 and 3 of

C. D. Zangmeister, J. G. Radney, E. P. Vicenzi, J. L. Weaver, Filtration Efficiencies of Nanoscale Aerosol by Cloth Mask Materials Used to Slow the Spread of SARS-CoV-2. ACS Nano. 14, 9188–9200 (2020). Link

Your comparison shouldn’t be detailed; note the shape of the curve, any extrema, and whether it depends on *N*. (Photos of fibers are in Figure 1, by the way.)

Here’s Figure 3B, filtration efficiency (FE) vs. particle size:

If you’ve done the exercise correctly, you should be amazed!

If you want to know more about the physics of filtration, see the reference cited in this paper. “Contact” involves a lot of neat physics, much of which is described in the book by Israelachvilli noted in the footnote. Our point here is simply to illustrate how dominant diffusion is in setting the overall behavior of this system, as is the case for nearly everything microscopic!

I thought it was a fun exercise!

### Today’s illustration…

A facemask, of course! It’s not a self-portrait; my colored pencil sketch is based on this photograph of some random guy.

— Raghuveer Parthasarathy, November 2, 2020

*** **[Footnote] van der Waals forces are very strong at short range, and are always attractive. For much more on this topic, see *Intermolecular and Surface Forces* by Jacob Israelachvili, an excellent book.